3/07/2020
We decided to do central Colombia, basically because it is where the capital is.
We built a model for the number of confirmed cases using all the others covariates (plus some we created) and we estimated the predictive accuracy of our selected model.
We decided to consider as central Colombia the following cities:
Here we can see our selected cities. The color of the pins is related with the number of cases: if they are less than \(10\) the color is “green”, if they are less than \(100\) the color is “orange”, otherwise it is “red”.
We transformed the Fecha de diagnóstico variable into a Date variable, we created a variable Grupo de edad, we transformed the variables Grupo de edad, Departamento o Distrito, Ciudad de ubicación, Sexo, Atención, Tipo into factors, we cleaned the column PaÃs de procedencia and created the variable Continente de procedencia.
## ID de caso Fecha de diagnóstico Ciudad de ubicación Departamento o Distrito ## 1 1 2020-03-06 Bogotá Bogotá D.C. ## 2 2 2020-03-09 Buga Valle del Cauca ## 3 3 2020-03-09 MedellÃn Antioquia ## 4 4 2020-03-11 MedellÃn Antioquia ## 5 5 2020-03-11 MedellÃn Antioquia ## 6 6 2020-03-11 Itagüà Antioquia ## Atención Edad Sexo Tipo PaÃs de procedencia Grupo de edad ## 1 Recuperado 19 F Importado Italia 19-30 ## 2 Recuperado 34 M Importado España 31-45 ## 3 Recuperado 50 F Importado España 46-60 ## 4 Recuperado 55 M Relacionado Colombia 46-60 ## 5 Recuperado 25 M Relacionado Colombia 19-30 ## 6 Casa 27 F Relacionado Colombia 19-30 ## Continente de procedencia ## 1 Europa ## 2 Europa ## 3 Europa ## 4 Colombia ## 5 Colombia ## 6 Colombia
Number of cases confirmed day by day
Here the growth seems exponential (and this is consistent with the fact that we are studying the early stages of the outbreak).
brks <- seq(-250, 250, 50)
lbls <- as.character(c(seq(-250, 0, 50), seq(50, 250, 50)))
ggplot(data=colombia_covid, aes(x=`Departamento o Distrito`, fill = Sexo)) +
geom_bar(data = subset(colombia_covid, Sexo == "F")) +
geom_bar(data = subset(colombia_covid, Sexo == "M"), aes(y=..count..*(-1))) +
scale_y_continuous(breaks = brks,
labels = lbls) +
coord_flip() +
labs(title="Spread of the desease across genders",
y = "Number of cases",
x = "Department",
fill = "Gender") +
theme_tufte() +
theme(plot.title = element_text(hjust = .5),
axis.ticks = element_blank()) +
scale_fill_brewer(palette = "Dark3")
#compute percentage so that we can label more precisely the pie chart
age_groups_pie <- colombia_covid %>%
group_by(`Grupo de edad`) %>%
count() %>%
ungroup() %>%
mutate(per=`n`/sum(`n`)) %>%
arrange(desc(`Grupo de edad`))
age_groups_pie$label <- scales::percent(age_groups_pie$per)
age_pie <- ggplot(age_groups_pie, aes(x = "", y = per, fill = factor(`Grupo de edad`))) +
geom_bar(stat="identity", width = 1) +
theme(axis.line = element_blank(),
plot.title = element_text(hjust=0.5)) +
labs(fill="Age groups",
x=NULL,
y=NULL,
title="Distribution of the desease across ages") +
coord_polar(theta = "y") +
#geom_text(aes(x=1, y = cumsum(per) - per/2, label=label))
geom_label_repel(aes(x=1, y=cumsum(per) - per/2, label=label), size=3, show.legend = F, nudge_x = 0) +
guides(fill = guide_legend(title = "Group"))
age_pie
theme_set(theme_classic())
ggplot(colombia_covid, aes(x = `Fecha de diagnóstico`)) +
scale_fill_brewer(palette = "Set3") +
geom_histogram(aes(fill=Tipo), width = 0.8, stat="count") +
theme(axis.text.x = element_text(angle=65, vjust=0.6)) +
labs(title = "Daily number of confirmed cases",
subtitle = "subdivided across type",
x = "Date of confirmation",
fill = "Type")
I think that en estudio means that it is not clear while the case is imported or not, however it seems like there are more imported cases, we can count them:
type_pie <- colombia_covid %>%
group_by(Tipo) %>%
count() %>%
ungroup() %>%
mutate(per=`n`/sum(`n`)) %>%
arrange(desc(Tipo))
type_pie$label <- scales::percent(type_pie$per)
type_pie<-type_pie[names(type_pie)!="per"]
colnames(type_pie)<-c("Tipo", "Total number", "Percentage")
type_pie
## # A tibble: 3 x 3 ## Tipo `Total number` Percentage ## <fct> <int> <chr> ## 1 Relacionado 291 29.3% ## 2 Importado 467 47.0% ## 3 En estudio 235 23.7%
Elapsed time as predictor## ## Call: ## glm(formula = `Cumulative cases` ~ `Elapsed time`, family = poisson, ## data = cases) ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -4.7629 -4.0644 -0.8324 1.5642 6.4039 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) 2.461377 0.052885 46.54 <2e-16 *** ## `Elapsed time` 0.169656 0.002346 72.32 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for poisson family taken to be 1) ## ## Null deviance: 7859.81 on 24 degrees of freedom ## Residual deviance: 280.62 on 23 degrees of freedom ## AIC: 446.83 ## ## Number of Fisher Scoring iterations: 4
## ## Call: ## glm(formula = `Cumulative cases/Department` ~ `Elapsed time`, ## family = poisson, data = cases_relev_dep) ## ## Deviance Residuals: ## Min 1Q Median 3Q Max ## -13.9994 -4.7902 -2.1236 0.8586 16.3553 ## ## Coefficients: ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) 0.982659 0.062709 15.67 <2e-16 *** ## `Elapsed time` 0.167306 0.002779 60.20 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## (Dispersion parameter for poisson family taken to be 1) ## ## Null deviance: 8732.6 on 82 degrees of freedom ## Residual deviance: 3855.1 on 81 degrees of freedom ## AIC: 4285.4 ## ## Number of Fisher Scoring iterations: 5
As a first attempt, we fit a simple Poisson regression:
\[ ln\lambda_i = \alpha + \beta\cdot elapsed\_time_i \\ y_i \sim \mathcal{Poisson}(\lambda_i) \]
with \(i = 1,\dots,83\), and \(y_i\) represents the number of cases.
## Running /usr/lib/R/bin/R CMD SHLIB foo.c
## gcc -std=gnu99 -I"/usr/share/R/include" -DNDEBUG -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/Rcpp/include/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/unsupported" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/BH/include" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/src/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/" -I"/home/angela/.R/x86_64-pc-linux-gnu-library/3.6/rstan/include" -DEIGEN_NO_DEBUG -D_REENTRANT -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -include stan/math/prim/mat/fun/Eigen.hpp -fpic -g -O2 -fdebug-prefix-map=/build/r-base-V28x5H/r-base-3.6.3=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g -c foo.c -o foo.o
## In file included from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Core:88:0,
## from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Dense:1,
## from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4,
## from <command-line>:0:
## /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:1: error: unknown type name ‘namespace’
## namespace Eigen {
## ^~~~~~~~~
## /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/src/Core/util/Macros.h:613:17: error: expected ‘=’, ‘,’, ‘;’, ‘asm’ or ‘__attribute__’ before ‘{’ token
## namespace Eigen {
## ^
## In file included from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Dense:1:0,
## from /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4,
## from <command-line>:0:
## /home/angela/.R/x86_64-pc-linux-gnu-library/3.6/RcppEigen/include/Eigen/Core:96:10: fatal error: complex: File o directory non esistente
## #include <complex>
## ^~~~~~~~~
## compilation terminated.
## /usr/lib/R/etc/Makeconf:168: recipe for target 'foo.o' failed
## make: *** [foo.o] Error 1
## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 1). ## Chain 1: ## Chain 1: Gradient evaluation took 2.1e-05 seconds ## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.21 seconds. ## Chain 1: Adjust your expectations accordingly! ## Chain 1: ## Chain 1: ## Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 1: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 1: ## Chain 1: Elapsed Time: 0.149009 seconds (Warm-up) ## Chain 1: 0.106585 seconds (Sampling) ## Chain 1: 0.255594 seconds (Total) ## Chain 1: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 2). ## Chain 2: ## Chain 2: Gradient evaluation took 1.4e-05 seconds ## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds. ## Chain 2: Adjust your expectations accordingly! ## Chain 2: ## Chain 2: ## Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 2: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 2: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 2: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 2: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 2: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 2: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 2: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 2: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 2: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 2: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 2: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 2: ## Chain 2: Elapsed Time: 0.148065 seconds (Warm-up) ## Chain 2: 0.113132 seconds (Sampling) ## Chain 2: 0.261197 seconds (Total) ## Chain 2: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 3). ## Chain 3: ## Chain 3: Gradient evaluation took 1.3e-05 seconds ## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds. ## Chain 3: Adjust your expectations accordingly! ## Chain 3: ## Chain 3: ## Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 3: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 3: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 3: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 3: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 3: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 3: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 3: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 3: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 3: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 3: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 3: ## Chain 3: Elapsed Time: 0.162958 seconds (Warm-up) ## Chain 3: 0.110986 seconds (Sampling) ## Chain 3: 0.273944 seconds (Total) ## Chain 3: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 4). ## Chain 4: ## Chain 4: Gradient evaluation took 1.7e-05 seconds ## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.17 seconds. ## Chain 4: Adjust your expectations accordingly! ## Chain 4: ## Chain 4: ## Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 4: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 4: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 4: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 4: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 4: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 4: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 4: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 4: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 4: ## Chain 4: Elapsed Time: 0.162306 seconds (Warm-up) ## Chain 4: 0.115285 seconds (Sampling) ## Chain 4: 0.277591 seconds (Total) ## Chain 4:
## Inference for Stan model: poisson_regression. ## 4 chains, each with iter=2000; warmup=1000; thin=1; ## post-warmup draws per chain=1000, total post-warmup draws=4000. ## ## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat ## alpha 0.18 0 0.13 -0.08 0.10 0.18 0.26 0.45 849 1.01 ## beta 0.12 0 0.01 0.10 0.11 0.12 0.12 0.13 843 1.00 ## ## Samples were drawn using NUTS(diag_e) at Thu Jul 2 02:45:18 2020. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split chains (at ## convergence, Rhat=1).
## Chain 1: ## Chain 1: Gradient evaluation took 1.5e-05 seconds ## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds. ## Chain 1: Adjust your expectations accordingly! ## Chain 1: ## Chain 1: ## Chain 1: ## Chain 1: Elapsed Time: 0.144022 seconds (Warm-up) ## Chain 1: 0.106378 seconds (Sampling) ## Chain 1: 0.2504 seconds (Total) ## Chain 1: ## Chain 2: ## Chain 2: Gradient evaluation took 1.4e-05 seconds ## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.14 seconds. ## Chain 2: Adjust your expectations accordingly! ## Chain 2: ## Chain 2: ## Chain 2: ## Chain 2: Elapsed Time: 0.158632 seconds (Warm-up) ## Chain 2: 0.1293 seconds (Sampling) ## Chain 2: 0.287932 seconds (Total) ## Chain 2: ## Chain 3: ## Chain 3: Gradient evaluation took 1.2e-05 seconds ## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds. ## Chain 3: Adjust your expectations accordingly! ## Chain 3: ## Chain 3: ## Chain 3: ## Chain 3: Elapsed Time: 0.159196 seconds (Warm-up) ## Chain 3: 0.132035 seconds (Sampling) ## Chain 3: 0.291231 seconds (Total) ## Chain 3: ## Chain 4: ## Chain 4: Gradient evaluation took 1.5e-05 seconds ## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds. ## Chain 4: Adjust your expectations accordingly! ## Chain 4: ## Chain 4: ## Chain 4: ## Chain 4: Elapsed Time: 0.151169 seconds (Warm-up) ## Chain 4: 0.109528 seconds (Sampling) ## Chain 4: 0.260697 seconds (Total) ## Chain 4:
Looking at Rhat we can see that we have reached the convergence.
theme_set(bayesplot::theme_default())
mcmc_scatter(as.matrix(fit.model.Poisson, pars=c("alpha", "beta") ), alpha=0.2)
Check the posterior:
y_rep <- as.matrix(fit.model.Poisson, pars="y_rep") ppc_dens_overlay(y = model.data$cases, y_rep[1:200,])
The model is not able to capture low and high numbers of new cases.
The fit is not satisfactory, it is probably due to overdispersion, we can check the residuals to confirm this hypothesis:
#in this way we check the standardized residuals mean_y_rep<-colMeans(y_rep) std_residual<-(model.data$cases - mean_y_rep) / sqrt(mean_y_rep) qplot(mean_y_rep, std_residual) + hline_at(2) + hline_at(-2)
The variance of the residuals increases as the predicted value increase. The standardized residuals should have mean 0 and standard deviation 1 (hence the lines at +2 and -2 indicates approximate 95% error bounds). The variance of the standardized residuals is much greater than 1, indicating a large amount of overdispersion.
Classically the problem of having overdispersed data is solved using the negative binomial model instead of the Poisson one.
## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 1). ## Chain 1: ## Chain 1: Gradient evaluation took 9e-06 seconds ## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds. ## Chain 1: Adjust your expectations accordingly! ## Chain 1: ## Chain 1: ## Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 1: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 1: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 1: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 1: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 1: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 1: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 1: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 1: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 1: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 1: ## Chain 1: Elapsed Time: 0.113289 seconds (Warm-up) ## Chain 1: 0.063062 seconds (Sampling) ## Chain 1: 0.176351 seconds (Total) ## Chain 1: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 2). ## Chain 2: ## Chain 2: Gradient evaluation took 7e-06 seconds ## Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds. ## Chain 2: Adjust your expectations accordingly! ## Chain 2: ## Chain 2: ## Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 2: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 2: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 2: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 2: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 2: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 2: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 2: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 2: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 2: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 2: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 2: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 2: ## Chain 2: Elapsed Time: 0.083492 seconds (Warm-up) ## Chain 2: 0.068111 seconds (Sampling) ## Chain 2: 0.151603 seconds (Total) ## Chain 2: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 3). ## Chain 3: ## Chain 3: Gradient evaluation took 7e-06 seconds ## Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds. ## Chain 3: Adjust your expectations accordingly! ## Chain 3: ## Chain 3: ## Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 3: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 3: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 3: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 3: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 3: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 3: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 3: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 3: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 3: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 3: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 3: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 3: ## Chain 3: Elapsed Time: 0.070711 seconds (Warm-up) ## Chain 3: 0.064096 seconds (Sampling) ## Chain 3: 0.134807 seconds (Total) ## Chain 3: ## ## SAMPLING FOR MODEL 'poisson_regression' NOW (CHAIN 4). ## Chain 4: ## Chain 4: Gradient evaluation took 7e-06 seconds ## Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds. ## Chain 4: Adjust your expectations accordingly! ## Chain 4: ## Chain 4: ## Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup) ## Chain 4: Iteration: 200 / 2000 [ 10%] (Warmup) ## Chain 4: Iteration: 400 / 2000 [ 20%] (Warmup) ## Chain 4: Iteration: 600 / 2000 [ 30%] (Warmup) ## Chain 4: Iteration: 800 / 2000 [ 40%] (Warmup) ## Chain 4: Iteration: 1000 / 2000 [ 50%] (Warmup) ## Chain 4: Iteration: 1001 / 2000 [ 50%] (Sampling) ## Chain 4: Iteration: 1200 / 2000 [ 60%] (Sampling) ## Chain 4: Iteration: 1400 / 2000 [ 70%] (Sampling) ## Chain 4: Iteration: 1600 / 2000 [ 80%] (Sampling) ## Chain 4: Iteration: 1800 / 2000 [ 90%] (Sampling) ## Chain 4: Iteration: 2000 / 2000 [100%] (Sampling) ## Chain 4: ## Chain 4: Elapsed Time: 0.089332 seconds (Warm-up) ## Chain 4: 0.066054 seconds (Sampling) ## Chain 4: 0.155386 seconds (Total) ## Chain 4:
## Inference for Stan model: poisson_regression. ## 4 chains, each with iter=2000; warmup=1000; thin=1; ## post-warmup draws per chain=1000, total post-warmup draws=4000. ## ## mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat ## alpha 2.46 0 0.05 2.35 2.42 2.46 2.49 2.56 536 1.01 ## beta 0.17 0 0.00 0.17 0.17 0.17 0.17 0.17 547 1.01 ## ## Samples were drawn using NUTS(diag_e) at Thu Jul 2 02:45:29 2020. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split chains (at ## convergence, Rhat=1).
